Economics of Inequality 

Thomas Piketty

Academic year 2012-2013

Course Notes : Models of Wealth Accumulation and Distribution

Summary findings:

(1) Pure life-cycle models generate too little wealth inequality

(2) Models with bequest and multiplicative wealth shocks are necessary to account for observed wealth concentration

(3) Models with explosive Pareto coefficients might be suitable to explain pre-WW1 inequality peak; they can also explain the multiplicative impact of progressive taxation



1. Pure life-cycle model: the Modigliani triangle formula


Pure life-cycle model = individuals die with zero wealth (no inheritance), wealth accumulation is entirely driven by life-cycle motives (i.e. savings for retirement)


Simplest model to make the point: fully stationary model n=g=r=0 (zero population growth, zero economic growth, zero interest rate = capital is a pure storage technology and has no productive use)

(see e.g. F. Modigliani, “Life Cycle, Individual Thrift and the Wealth of Nations”, AER 1986)


Age profile of labor income:


Note YLa = labor income at age a


YLct= YLa =YL>0 for all A<a<A+N 

YLct= YLa = 0 for all A+N<a<A+D (& 0<a<A)


with c = cohort (birth year), t = current year, a=t-c=current age


I.e. people become adult and start working at age A, work during N years, retire at age A+N, and die at age A+D: labor length = N, retirement length = D-N


(say: A=20, A+N=60, A+D=70, i.e. N=40, D-N=10)


(stationary model: drop year t)


Per capita (adult) national income Y = NYL/D


Preferences: full consumption smoothing

(say, U = ∑A<a<A+D U(Ca)/(1+θ)a, with θ=0)


Everybody fully smoothes consumption to C=NYL/D (= per capita output Y)

In order to achieve this they save during labor time and dissave during retirement time


Note Sa= savings (= Ya – C) (=YLa – C)


We have:


Sa=(1-N/D)YL for all A<=a<=A+N,

Sa=-NYL/D for all A+N<=a<=A+D


I.e. during retirement, consumption fully comes from consuming past accumulation


Note Wa= wealth at age a


We get the following wealth accumulation equation:


Wa=(a-A)(1-N/D)YL for all A<a<A+N 

Wa= N(1-N/D)YL-(a-A-N)NYL/D for all A+N<a<A+D


>>> “hump-shaped” (inverted-U) age-wealth profile, Wa back to 0 for a=A+D


Average wealth W = N/D x  N(1-N/D)YL/2 + (D-N)/D x N(1-N/D)YL/2

i.e. average W = (D-N)Y/2


Proposition: Aggregate wealth/income ratio W/Y = (D-N)/2 = half of retirement length = “Modigliani triangle formula”


E.g. if retirement length D-N = 10 years, then W/Y = 500%

(and if D-N = 20 years, then W/Y=1000%…)


Lessons from Modigliani triangle formula:

(1) pure life-cycle motives (no bequest) can generate large and reasonable wealth/income ratios

(2) aggregate wealth/income ratio is independant of income level and solely depends on demographics (previous authors had to introduce relative income concerns in order to avoid higher savings and accumulation in richer economies);


Note that in this stationary model, aggregate savings = 0: i.e. at every point in time positive savings of workers are exactly offset by negative savings of retirees; but this is simply a trivial consequence of stationnarity: with constant capital stock, no room for positive steady-state savings


Extension to population growth n>0 : then the savings rate s is >0: this is because younger cohorts (who save) are more numerous than the older cohorts (who dissave)

Check: with population growth at rate n>0,

proportion of workers in the adult population = (1-exp(-nN))/(1-exp(-nD)) (> N/D for n>0)

I.e. s(n) = 1 - (N/D)/ [(1-exp(-nN))/(1-exp(-nD))] >0

Put numbers: in practice this generates savings rates that are not so small, e.g. for n=1% this gives s=4,5% for D-N=10yrs retirement, and s=8,8% for D-N=20yrs retirement (keeping N=40yrs)

Wealth accumulation:  W/Y = s/n = s(n)/n

i.e. wealth/income ratio = savings rate/population growth

(dWt/dt = sYt and Wt = βYt >> β = s/n)

>> for n=0, W/Y = (D-N)/2; for n>0, W/Y < (D-N)/2; i.e. W/Y rises with retirement length D-N, but declines with population growth n

Put numbers: W/Y=452% instead of 500% for N=40,D-N=10,n=1%.

Intuition: with larger young cohorts (who have wealth close to zero), aggregate wealth accumulation is smaller; mathematically, s(n) rises with n, but less than proportionally; of course things would be reversed if N was small as compared to D, i.e. if young cohorts were reaching their accumulation peak very quickly

(also, this result depends upon the structure of population growth: aging-based population growth generates a positive relationship between population growth and wealth/income ratio, unlike in the case of generational population growth)


Extension to economic growth g>0 : then s>0 for the same reasons as the population growth: young cohorts are not more numerous, but they are richer (they have higher lifetime labor income), so they save more than the old dissave


Extension to positive capital return r>0 : other things equal, the young need to save less for their old days (thanks to the capital income YK=rW; i.e. now Y=YL+YK);  if n=g=0 but r>0, then one can easily see that aggregate consumption C is higher than aggregate labor income YL, i.e. aggregate savings are smaller than aggregate capital income, i.e. S<YK, i.e. savings rate s=S/Y < capital share α = YK/Y

(s<α = typically what we observe in practice, at least in countries with n+g small)

Main limitations of the lifecycle model: it generates too little wealth inequality. I.e. in the lifecycle model, wealth distribution is simply the mirror image of income distribution - while in practice wealth distribution is always a lot more unequal than income distribution.


Obvious culprit: the existence of inherited wealth and of multiplicative, cumulated effects induced by wealth transmission over time and across generations. This can naturally generate much higher wealth concentration.


2. Pure dynastic model

Pure dynastic model = individuals maximize dynastic utility functions, as if they were infinitely lived; death is irrelevant in their wealth trajectory, so that they die with positive wealth, unlike in the lifecycle model:


See e.g. Blanchard-Fisher, Lectures on macroeconomics, Chapter 2, MIT Press, 1989

G. Bertola, R. Foellmi & J. Zweimuller, Income distribution in macroeconomic models, Chapter 3, Princeton University Press, 2006


Dynastic utility function:


Ut = ∑t≥0 U(ct)/(1+θ)t

(U’(c)>0, U’’(c)<0)


Infinite-horizon, discrete-time economy with a continuum [0;1] of dynasties.


For simplicity, assume a two-point distribution of wealth. Dynasties can be of one of two types: either they own a large capital stock ktA, or they own a low capital stock ktB (ktA > ktB). The proportion of high-wealth dynasties is exogenous and equal to λ (and the proportion of low-wealth dynasties is equal to 1-λ), so that the average capital stock in the economy kt is given by:


kt = λktA + (1-λ)ktB


Output per labor unit is given by a standard production function f(kt) (f’(k)>0, f’’(k)<0), where kt is the average capital stock per capita of the economy at period t. Markets for labor and capital are assumed to be fully competitive, so that the interest rate rt and wage rate vt are always equal to the marginal products of capital and labor:


rt = f’(kt)

vt = f(kt) - rtkt


Proposition: (1) In long-run steady-state, the interest rate r* and the average capital stock k* are uniquely determined by the utility function and the technology (irrespective of initial conditions): in steady-state, r* is necessarily equal to θ, and k* must be such that: f’(k*)=r*=θ (“golden rule of capital accumulation”)

(2) Any distribution of wealth (kA , kB) such as average wealth = k* is a steady-state


The result comes directly from the first-order condition:


U’(ct)/ U’(ct+1) = (1+rt)/(1+θ)


I.e. if the interest rate rt is above the rate of time preference θ, then agents choose to accumulate capital and to postpone their consumption indefinitely (ct<ct+1<ct+2<…) and this cannot be a steady-state. Conversely, if the interest rate rt is below the rate of time preference θ, agents choose to desaccumulate capital (i.e. to borrow) indefinitely and to consume more today (ct>ct+1>ct+2>…). This cannot be a steady-state either.


Note: if f(k) = kα (Cobb-Douglas), then long run β = k/y = α/r


Note: in steady-state, s=0 (zero growth, zero savings)


Average income: y = v + rk* = f(k) = average consumption

High-wealth dynasties: income yA = v + rkA  (=consumption)

Low-wealth dynasties: income yB = v + rkB  (=consumption)


>>> everybody works the same, but some dynasties are permanently richer and consume more


Dynastic model = completely different picture of wealth accumulation than life-cycle model

In the pure dynastic model, wealth accumulation = class war

In the pure lifecycle model, wealth accumulation = age war


In pure dynastic model, any wealth inequality is self-sustaining

In pure lifecycle model, zero wealth inequality

Dynastic model with positive (exogenous) productivity growth g:

Modified Golden rule: r* = θ + σg

With: 1/σ = intertemporal elasticity of substitution: U(C) = C1-σ/(1-σ)

In steady-state, sL=0, but sK=g/r, i.e. C = YL + (r-g)K, i.e. dynasties save a fraction g/r of their capital income (and consume the rest), so that their capital stock grows at rate g, i.e. at the same rate as labor productivity and output


Main limitations of dynastic model:

(1) Any level of wealth inequality can be a steady-state (complete indeterminacy) (=complete opposite to lifecycle model, where wealth inequality entirely determined by income inequality)

(2) Infinite horizon implies infinite long run elasticity of savings with respect to net-of-tax rate of return (i.e. net-of-tax rate of return must be exactly equal to r* in the long run); this is pretty extreme; this trivially implies zero optimal capital taxes (including from the viewpoint of zero wealth dynasties); i.e. same conclusion as in lifecycle model, but for completely different reasons


3. Middle case: wealth-in-the-utility, finite-horizon models


Each agent i is now assumed to maximize a utility function of the following form:


Vi = V( UCi , wi(D) )                


With: UCi = [ ∫A≤a≤D e-θ(a-A) ci(a)1-σda ]1/(1-σ) = utility from lifetime consumption

(i.e. between age a=A=adulthood and age a=D=death, say A=20, D=80)

w(D) =end-of-life wealth = bequest for next generation

V(U,w) = (1-s)log(U)+slog(w) (or equivalently V(U,w) = U1-s ws )

s = share of lifetime resources devoted to end-of-life wealth wi(D)

1-s = share of lifetime resources devoted to lifetime consumption


Multiples interpretations: utility for bequest, direct utility for wealth, reduced form for precautionary savings, etc.


If s goes to zero, then we’re back to the pure lifecycle model

If D goes to infinity, then we’re back to the pure dynastic model


→ this model is more flexible and realistic


For more details & additional references, see Piketty 2010, section 5 & Appendix E


4. Where Do We Stand Between Pure Lifecycle and Pure Dynastic Model?

See Piketty, "On the Long Run Evolution of Inheritance", WP 2010, QJE 2011 

See Presentation slides for a summary of main results

→ in 1950s-1960s (Modigliani), inheritance was indeed very low; but inheritance flows are now almost back to their pre-WW1 level

→ with g low and r>g, wealth coming from the past is being capitalized faster than national income, so (for large classes of preferences) inheritance flows are bound to be large

(of course with pure lifecycle preferences, agents always choose to die with zero wealth, so inheritance flows = zero even if g low and r>g; but the point is that real-world agents clearly have non-lifecycle reasons for accumulating wealth: bequest, prestige, security motives seem to matter; even in 1950s-60s, ratio between average wealth of decedents and average wealth of the living is at least 100%, not 0%; taking as given such preferences, then g low and r>g imply large inheritance flows)

5. Why are wealth distributions so unequal? Multiplicative models & Pareto distributions


Many unequalizing forces and shocks: tastes, returns, demographic shocks, etc.

(+ credit market imperfections: if borrowing rate > rental rate, then tenant dynasties keep paying rents to landlord dynasties)

Here we make a simple point: with inheritance, multiplicative wealth shocks naturally generate large steady-state wealth concentration (Pareto distributions), and even more so if g low and r>g.

I.e. r>g matters not only for the aggregate level of inheritance, but also for the inequality of inheritance

Simplified version of the wealth-in-the-utility, finite-horizon model:

- each generation lives exactly one period

- each individual i in generation t receives net-of-labor-tax labor income (1-τL)yLt + net-of-bequest-tax capitalized bequest (1-τB)erHbti , and maximizes Vti(c,w)=(1-sti)log(c)+stilog(w) (or equivalently Vi(c,w)=cstiw1-sti) in order to allocate his total ressources between consumption cti and end-of-life wealth wti

→ Individual-level transition equation:

bt+1i=(1-τL)stiyLt + (1-τB)stierHbti

With: bt+1i=wti=bequest received by generation t+1 (each individual i has exactly one children)

erH = capitalization factor (generational rate of return)

r = annual rate of return; H = generation length

(e.g. if r=4%, H=30 years, then erH=3.32)

sti = taste-for-wealth or taste-for-bequest parameter

yLti=yLt = for simplicity, all individuals are assumed to have the same labor income

yLt=(1-α)yt = labor income is a fixed fraction 1-α of per capita output yt

yt=yoegHt = per capita output grows at generational rate egH

g=annual productivity growth rate (exogenous)

(e.g. if g=2%, H=30 years, egH=1.82)

(see Piketty-Saez 2012 for more general versions of this model and for technical details)

Binomial random tastes for wealth (i.i.d. each generation):

sti=s*>0 with proba p>0 ("wealth-lovers")

sti=0 with proba 1-p ("consumption-lovers")

Note: one could replace taste shocks by demographic shocks (age at parenthood, age at death, number of children, rank of children) or shocks to rates of return; all these shocks matter a lot in practice; the equations would be the same with all these shocks

s* = generational saving rate of wealth-lovers

µ* = (1-τB)s*e(r-g)H = generational wealth reproduction rate of wealth-lovers

If sti=0 , then bt+1i=0

→ children with consumption-loving parents receive no bequest

If sti=s*, then bt+1i=(1-τL)s*yLt + µ* egH bti

→ children with wealth-loving parents receive positive bequests growing at rate µ* (above the generational growth rate of the economy egH) across generations

→ after many successive generations with wealth-loving parents (or more generally with high demographic or returns shocks), inherited wealth can be very large

s = ps* = aggregate generational saving rate

µ = (1-τB)se(r-g)H =pµ*= aggregate generational wealth reproduction rate

Non-explosive aggregate path: µ<1

Non-explosive aggregate path with unbounded distribution of normalized inheritance: µ<1<µ*=µ/p

Aggregate transition equation for average bequest bt :

bt+1=(1-τL)syLt + µ egH bt

By dividing both sides of the equation by yt+1=egH yt , and by defining byt=erHbt/yt the capitalized inheritance/output ratio, we get:

byt+1= (1-τL)(1-α)se(r-g)H + µbyt

I.e.  byt→ by=(1-τL)(1-α)se(r-g)H/(1-µ)

Result 1: The aggregate inheritance/output ratio is an increasing function of r-g

We can now compute the steady-state distribution φ(z) of normalized inheritance zti=bti/bt.

As t→+∞, the transition equation for zti looks as follows:

If sti=0, then zt+1i=0

If sti=s*, then zt+1i=(1-τL)(1-α)s*e(r-g)H/by + (µ/p) zti

(divide both sides of bt+1i=(1-τL)s*yLt + (µ/p) egH bti by bt+1, and note that as t→+∞, bt+1≈bye-rHyt )

I.e. if sti=s*, then zt+1i=(1-µ)/p+ (µ/p) zti

Therefore the steady-state distribution φ(z) looks as follows:

z=z0=0 with proba 1-p (children with zero-wealth-taste parents)

z=z1=(1-µ)/p with proba (1-p)p (children with wealth-loving parents but zero-wealth-taste grand-parents)


z=zk+1=(1-µ)/p + (µ/p)zk > zk with proba (1-p)pk+1 (children with wealth-loving ancesters during the past k+1 generations)

We have:

zk = [(1-µ)/(µ-p)] [ (µ/p)k - 1 ] ≈ [(1-µ)/(µ-p)] (µ/p)k  as k→+∞

1-Φ(zk) = proba(z≥zk) = ∑k'>k (1-p)pk' = pk

That is, as z→+∞, log[1-Φ(z)] ≈ αP ( log[ω] - log[z] ), i.e. 1-Φ(z) ≈ (ω/z)αP 

With ω = (1-µ)/(µ-p)

and: Pareto coefficient αP = log[1/p]/log[µ/p] >1

and inverted Pareto coefficient βP=αP/(1-αP) >1

For given p:

As µ→1 , αP→1 and βP→+ (infinite inequality): an increase in µ=(1-τB)ps*e(r-g)H means a larger wealth reproduction rate µ* for wealth-lovers, i.e. a stronger amplification of inequality

(conversely, as µ→p , αP→+∞ and βP→1 (zero inequality) (if µ<p, i.e. µ*<1, then z is bounded above: zero inequality at the top)


For given µ:

As p→0 , αP→1 and βP→+ (infinite inequality) (a vanishingly small fraction of the population gets an infinitely large shock)

(conversely, as p→µ , αP→+∞ and βP→1 (zero inequality) (if p>µ, i.e. µ*<1, then z is bounded above)



Result 2: The inequality of inheritance is an increasing function of r-g

Note 1. If bequest tax rate τB0% (France 1900-1910), and saving rate s* rises strongly with income level (consumption satiation effect), then the asymptotic µ is >1 for high wealth levels → explosive concentration of wealth

Note 2. Conversely, bequest taxes - and more generally capital taxes - reduce the rate of wealth reproduction, and therefore reduce steady-state wealth concentration.

Note 3. The same ideas and formulas for Pareto coefficients work for different kinds of shocks.

- Primogeniture: shock = rank at birth; the richest individuals are the first born sons of first born sons of first born sons etc.; see Stiglitz Econometrica 1969 

- Family size: shock = number of siblings; the richest individuals are the single children of single children of single children etc.; see Cowell 1998 (more complicated formula

- Rates of return: shock = rti instead of sti = exactly the same mutiplicative wealth process as with taste shocks → Pareto upper tails in the limit, see e.g. Benhabib-Bisin-Zhu 2011, 2013, Nirei 2009

Note 4. With primogeniture (binomial shock), the formula is exactly the same. See e.g. Atkinson-Harrison 1978 p.213 (referred to in Atkinson-Piketty-Saez 2011 p.58), who generalize the Stiglitz 1969 formula and get: αP = log(1+n)/log(1+(1-t)sr). This is the same formula as αP = log[1/p]/log[µ*]: 1+n = population growth, so probability that a good shock occurs - i.e. being the eldest son = 1/(1+n) = p; 1+(1-t)sr = net-of-tax reproduction rate in case a good shock occurs = µ*.

Note 5. The Cowell 1998 result is more complicated because families with many children do not return to zero (unless infinite number of children), so there is no closed form formula for the Pareto coefficient α, which must solve the following equation: ∑ (pkk/2) (2µ/k)α= 1, where pk= fraction of parents who have k children, with k=1,2,3,etc., and µ = average generational rate of wealth reproduction.

Note 6. More generally, one can show that for any random multiplicative process zt+1i= µtiztiti , where µti= i.i.d. multiplicative shock with mean µ=E(µti)<1, εti=additive shock (possibly random), then the steady-state distribution has a Pareto upper tail with coefficient α, which must solve the following equation: E(µtiα)=1 (see Nirei 2009, p.9). Special case: p (µ/p)α=1, i.e. α=log(1/p)/log(µ/p). More generally, as long as µti>1 with some positive probability, there exists a unique α>1 s.t. E(µtiα)=1. One can easily see that for a given average µ=E(µti)<1, α→1 if the variance of shocks goes to infinity (and α→∞ if the variance goes to zero). 

Note 7. For attempts to test Pareto coefficient formulas using wealth distribution data, see Dell JEEA 2005 (Germany vs France historical comparison: can lower bequest taxes explain higher wealth concentration in Germany?) and Fiaschi-Marsili 2009 (US and Italy: can the decline in capital income tax rates between the 1980s and the 2000s explain the decline in observed Pareto coefficients, i.e. rise in wealth concentration?) → this is imperfect, but this is the way to go